Applied Mathematics and Computation 168 (2005) 1079–1085

www.elsevier.com/locate/amc

An approach to VaR for capital markets with Gaussian mixture Ming-Heng Zhang a

a,*

, Qian-Sheng Cheng

b

School of Economics, Shanghai University of Finance and Economics, No. 777, Guo-Ding Road, Shanghai 200433, China b School of Mathematical Sciences, Peking University, Beijing 100871, China

Abstract An approach to VaR (value-at-risk) for capital markets is proposed with Gaussian mixture. Considering the impacts of the components in a Gaussian mixture, an approach to VaR for capital markets is proposed to describe risk structure in capital markets. This approach can be programmed in parallel. Empirical computation of VaR for China securities markets and the Forex markets are provided to demonstrate the proposed method. Ó 2004 Published by Elsevier Inc. Keywords: Gaussian mixture; Value-at-risk; VaR; Parallel Computation

1. Introduction The computational framework for value-at-risk (VaR) is a useful methodology for estimating the exposure of a given portfolio of securities to different *

Corresponding author. Tel.: +86 21 6590 4469/3687; fax: +86 21 6590 3687. E-mail addresses: [email protected] (M.-H. Zhang), [email protected] (Q.-S. Cheng). 0096-3003/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.10.004

1080

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

kinds of risk inherent in financial environment. One driving force behind the popularity of this technique is the release to the studies of many researchers and the documents of JP Morgan and Basle Committee on Banking Supervision [1,2]. Another one is so-called random walks hypothesis about capital markets [3]. However, empirical evidence showed that the returns are actually fat tailed and that there exists non-random walks in capital markets and thus suggested that the assumption that returns of financial assets are normally distributed is inappropriate [4,5]. VaR calculated under the normal assumption underestimates the actual risk [6–9]. Zangari treated two components in a Gaussian mixture to fit the fat of financial assets [6]. Venkataraman provided an estimation techniques for value-at-risk in a mixture of normal distributions [7]. Hull and White suggested to use alternative distributions, such as a mixture of two normal distributions, to model the return of financial assets [8]. Li made use of statistics such as volatility, skewness and kurtosis to capture the extreme tail [9]. It is seen that these methods paid attention to few components in a Gaussian mixture. However, empirical studies showed that the number of components is greater than two in a Gaussian mixture for the returns of financial assets [10–12]. In this paper, an approach to VaR for capital markets with Gaussian mixture takes into account of not only non-normal distribution but also the impacts of the components in a Gaussian mixture. In Section 2, we propose an approach to VaR for capital markets with Gaussian mixture that uses lots of the components in a Gaussian mixture to describe risk structure in capital markets. In Section 3, we provide empirical computation of VaR for China securities markets and the Forex markets to demonstrate the proposed method. In Section 4, some discussions are given.

2. The computational framework for value-at-risk In order to compute VaR for capital markets on the condition of non-random walks, we use Gaussian mixture to measure the risk of financial assets in capital markets. It is well known that Gaussian mixture can be used not only to fit returns of financial assets but also to capture non-normality of financial assets movements [11,12]. So, we pay attention to computational framework of VaR for capital markets with Gaussian mixture. Let S t 2 R be the financial asset prices series (stock, index, or exchange rate) and S ¼ fS 1 ; S 0 ; S 1 ; . . . ; S n1 g. Define returns of financial assets time series as rt ¼ logðS t =S t1 Þ

8S t 2 S 8t 2 f0; . . . ; n  1g

ð1Þ

and let R ¼ fx0 ; x1 ; . . . ; xn1 g and S1 = S0. Suppose that a component distribution fk ðrjlk ; r2k Þ is Gaussian, i.e. N ðlk ; r2k Þ, named this component as xk. Let K be the number of components,

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

1081

flk ; r2k gKk¼1 be the set of unknown parameters, and pk be mixing proportion of the kth component xk. Gaussian mixture can be expressed as K X f ðrjHÞ ¼ pk fk ðrjlk ; r2k Þ; ð2Þ k¼1

by a sum of distributions or equivalently be denoted as 8 rt jx1 N ðl1 ; r21 Þ; pðg ¼ x1 Þ ¼ p1 ; > > > > > > <... ð3Þ ðrt ; gÞ ¼ rt jxk N ðlk ; r2k Þ; pðg ¼ xk Þ ¼ pk ; > > > > . . . > > : rt jxK N ðlK ; r2K Þ; pðg ¼ xK Þ ¼ pK ; PK by a pair of random variables, where K P 1, 0 < pk 6 1, k¼1 p k ¼ 1 and H ¼ fp1 ; l1 ; r21 ; . . . ; pK ; lK ; r2K g and g is a latent indicator, i.e. g = xk if and only if rt 2 xk, with certain probabilistic structure. The unknown parameters, H and K, can be estimated by the issues [12–14]. The Empirical evidence has to show that the number of components, K, in a Gaussian mixture is greater than two [10–12] and conduces to an approach to VaR for capital markets with Gaussian mixture. This approach differs from the methods suggested by the researchers [6–9] in the impacts of components in a Gaussian mixture. It is the impacts of components in a Gaussian mixture that bring on various risky levels and make lots of the components to describe risk structure in capital markets. General speaking, VaR is a single, summary and statistical measure of possible portfolio profit and losses due to random walks in capital markets, which corresponds to the low a quantile of a distribution of returns. For any significance level a 2 (0,1), a distribution function F(r) for random variable r, VaR at the probability level a is defined as VaR ¼ F 1 ðaÞ ¼ inffrjF ðrÞ P ag;

ð4Þ

where, F(r) is Gaussian distribution. It is this so-called normality that stirs up a deviation of risk measurement from the real world. Considering the impacts of the components in a Gaussian mixture, an approach to VaR for capital markets is proposed. According to Gaussian mixture and VaR correspondingly to Eqs. (2)–(4), it is obvious that the following theorems hold true. Theorem 1. For the given significance level a 2 (0, 1), the respective VaRs of a Gaussian mixture and its kth component can be expressed as 8P R < K p VaRmx ðaÞ f ðrjl ; r2 Þ dr ¼ a; k k k¼1 k 1 ð5Þ : R VaRkmx ðaÞ f ðrjl ; r2 Þ dr ¼ a: 1

k

k

1082

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

The relationship to the various latent risks of a financial asset min fVaRkmx ðaÞg 6 VaRmx ðaÞ 6 max fVaRkmx ðaÞg

16k6K

16k6K

ð6Þ

always holds true. The VaRmx(a) and VaRkmx ðaÞ calculated according with Eq. (5) can describe the movement of price behaviors in capital markets at the given significance level a. Theorem 2. For the given VaR, the respective significance levels on a Gaussian mixture and its kth component can be represented as 8 K R > < amx ¼ P p VaR f ðrjl ; r2 Þ dr; k k 1 k ð7Þ k¼1 > R : k VaR 2 amx ¼ 1 f ðrjlk ; rk Þ dr: The relationship to the various latent confidences in a financial asset min fakmx g 6 amx ¼

16k6K

K X k¼1

pk akmx 6 max fakmx g 16k6K

ð8Þ

always keeps true. The amx and akmx evaluated according with Eq. (7) can feel the assurance on capital markets at the given VaR. The sum and substance of Theorems 1 and 2 can be also stated that for financial asset rt, which belongs to the kth components xk in a Gaussian mixture, its VaR follows that 8 VaR1mx ðaÞ; pðg ¼ x1 Þ ¼ p1 ; > > > > > > > < ðVaRðrt ; aÞ; gÞ ¼ VaRkmx ðaÞ; pðg ¼ xk Þ ¼ pk ; > > > > > > > : VaRKmx ðaÞ; pðg ¼ xK Þ ¼ pK ;

ð9Þ

where g is a latent indicator, i.e., g = xk if and only if rt 2 xk, with certain probabilistic structure fpk gKk¼1 and a is significance level. As a result, such things as a Gaussian mixture, Eqs. (2) and (3), Theorems 1 and 2 constitute an approach to VaR for capital markets that use lots of the components in a Gaussian mixture to describe the risk structure in capital markets as shown in Table 1 and Eq. ( 9).

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

1083

Table 1 An approach to VaR for capital markets at the behaviors of price movements and the psychologies of investors Measurement VaR for given a

a for given VaR

Computational formula min16k6K fVaRkmx ðaÞg

Under-estimated, Approximative-estimated, VaRmx(a) Over-estimated, max16k6K fVaRkmx ðaÞg Under-estimated, min16k6K fakmx g P Approximative-estimated, amx ¼ Kk¼1 pk akmx k Over-estimated, max16k6K famx g

Characteristic Adventurous Safety Craven Emotional Rational Depressed

3. Empirical computation In order to demonstrate the proposed method, we provide the empirical computation of VaR for China securities markets, i.e., Shanghai index and Shenzhen index, and the Forex markets, i.e., Interbank Rate of US dollar to Deutsche Mark, from 1996 to 2001. The unknown parameters, H and K, are estimated by the EM algorithm based on KS test. The tests showed that the number of components in a Gaussian mixture is more greater than two [12,14]. Based on Gaussian mixture, the behaviors of price movement and the psychologies of investors in capital market are showed in Table 2. With regard to the behaviors of price movement in capital markets, it is obvious that for the given significance level a = 0.99, VaRs on the condition of normal distribution, i.e., VaR(Shanghai) = 4.57109, VaR(Shenzhen) = 5.14056 and VaR(US2DEM) = 1.18379, differ from these in a principal component [12], i.e., VaR1mx ðShanghaiÞ ¼ 2:34484, VaR1mx ðShenzhenÞ ¼ 2:781 and VaR1mx ðUS2DEMÞ ¼ 1:91533, but also those in a Gaussian mixture respectively, i.e., VaRmx(Shanghai) = 5.63919, VaRmx(Shenzhen) = 6.04621 and VaRmx(US2DEM) = 1.44815. The differentiae show that VaR on the normal assumption is biased from the real world in contrast to a Gaussian mixture and then the behaviors of price movements in China securities markets were more risky than in the Forex markets, which resulted from the emerging markets in China. Referring to the psychologies of investors in capital markets it also be apparent that for the given VaRs on the condition of normal assumption, i.e., VaR(Shanghai) = 4.57109, VaR(Shenzhen) = 5.14056, and VaR(US2DEM) = 1.18379, the significance level a = 0.9900 in the Gaussian distribution is different from one in a principal component, i.e., a1mx ðShangahiÞ ¼ 0:999967, a1mx ðShenzhenÞ ¼ 0:999967 and a1mx ðUS2DEMÞ ¼ 0:921415, and another one in a Gaussian mixture respectively, i.e., amx(Shangahi) = 0.981153, amx(Shenzhen) = 0.981471 and amx(US2DEM) = 0.966648.

1084

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

Table 2 China Securities Markets: Daily Closing Composite Index, April 4, 1996 to April 2, 2001, respectively given a = 0.99 or VaR(Shanghai) = 4.57109 and VaR(Shenzhen) = 5.14056; the Forex Markets: Daily Averages of InterBank Rate, April 3, 1996 to April 3, 2001 respectively given a = 0.99 or VaR( USD2DEM) = 1.18379 r2k

K

pk

KShanghai = 1 KI Shanghai ¼ 5

1.000 1.09619e  1 3.67422e + 0 0.727 1.56275e  1 8.84165e  1 0.027 2.76539e + 0 2.26618e  1 0.201 3.00161e  1 1.04854e + 1 0.007 8.40024e + 0 1.73799e + 0 0.038 2.08946e + 0 2.39546e  1 VaRmx = 5.63919; amx=0.981153 1.000 1.30505e  1 4.63334e + 0 0.406 5.50217e  1 9.18597e  1 0.130 1.57799e + 0 9.02749e  1 0.160 1.71667e + 0 4.14210e + 0 0.137 7.33369e  1 1.76436e + 1 0.015 7.13784e  1 5.61976e  4 0.152 4.77083e  1 2.51223e  1 VaRmx = 6.04621; amx = 0.981471 1.000 2.25969e  2 2.48894e  1 0.494 4.97026e  2 2.46276e  1 0.215 1.97934e  2 1.06250e  2 0.136 1.68377e  1 3.64373e  2 0.155 1.33538e  1 7.28639e  1 VaRmx = 1.76659; amx = 0.951648

KShenzhen = 1 KI Shenzhen ¼ 6

KUSD2DEM = 1 KI USD2DEM ¼ 4

lk

VaR

a

4.57109 2.34484 3.8734 7.83698 5.33179 0.950491

0.9900 0.999967 0.999894 0.906376 0.999968 0.999968

5.14056 2.781000 0.63347 6.4537 9.04326 0.768961 0.689526

0.9900 0.999967 0.999968 0.953715 0.918974 0.999968 0.999968

1.18379 1.91533 0.220123 0.612669 1.85325

0.9900 0.921415 0.999968 0.999968 0.938583

It seems that the significance level a on the condition of normal assumption is unreliable in comparison of Gaussian mixture and then the psychologies of investors in China securities markets were more hazardous than in the Forex markets, which stemmed from irrational investors in China securities markets.

4. Discussion On above the statement and the empirical computation, it can be seen that this approach to VaR for capital markets with Gaussian mixture has advantage of the description of various risk levels. Illustrated by our empirical computation, an approach to VaR for capital markets with Gaussian mixture is different not only from one with the kth component in a Gaussian mixture but also from another one with Gaussian distribution and it seems that there will exist mistake without of Gaussian mixture. It should be pointed out that there exists a key problem to be resolved that the component in a Gaussian mixture that returns of financial assets in trading day belongs to must be determined in practice.

M.-H. Zhang, Q.-S. Cheng / Appl. Math. Comput. 168 (2005) 1079–1085

1085

References [1] J.P. Morgan, RiskMetricsTM, third edition, 1995. [2] J. Bessis, Risk Management in Banking, Wiley, New York, 1998. [3] E.F. Fama, Efficient capital markets: a review of theory and emperical work, Journal of Finance 25 (1970) 383–417. [4] A.W. Lo, A.C. MacKinlay, Stock market prices do not follow random walks: evidence from a simple specification test, Review of Financial Studies 1 (1988) 41–66. [5] J.L. Les´kow, The impact of stationarity assesment on studies of volatility and value-at-risk, Mathematical and Computer Modelling 34 (2001) 1213–1222. [6] P. Zangari, An improved methodology for measuring VaR, Risk Metrics Monitor, 2nd quarter, Reuters/J.P. Morgan, 7–25 (1996). [7] S. Venkataraman, Value at risk for a mixture of normal distributions: the use of quasiBayesian estimation techniques, Economic Perspectives, Federal Reserve Bank of Chicago (March) (1997) 2–13. [8] J. Hull, A. White, Value at risk when daily changes in market variables are not normally distributed, Journal of Derivatives 5 (1998) 9–19. [9] D.X.Li, Value at risk based on the volatility, skewness and kurtosis, Technical report, Riskmetrics Group, 1999. [10] S.J. Kon, Models of stock returns: a comparison, Journal of Finance 39 (1984) 47–165. [11] D. Kim, S.J. Kon, Alternative models for the conditional heteroscedasticity of stock returns, Journal of Business 67 (1994) 563–599. [12] M.H. Zhang, Q.S. Cheng, Gaussian mixture modelling to detect random walks in capital markets, Mathematical and Computer Modelling 38 (2003) 503–508. [13] G.J. McLanchlan, T. Krishnan, The EM Algorithm and Extensions, Wiley, New York, 1997. [14] M.H. Zhang, Q.S. Cheng, Determine the number of components in a mixture mode by the extended KS test, Patter Recognition Letters 25 (2004) 211–266.